How can NASA physicist and climatologist James E. Hansen, writing in the New York Times today, “say with high confidence” that recent heat waves in Texas and Russia “were not natural events” but actually “caused by human-induced climate change”?

It wasn’t all that long ago that respected MIT atmospheric scientist Kerry Emanuel flatly refuted the notion that you can pinpoint global warming as the cause of an extreme weather event. “It’s statistical nonsense,” he told PBS.

In 2005, Emanuel reported that hurricane intensity, which is fed by warmth, had increased some 80 percent during the previous 50 years, a period during which temperatures had increased more dramatically than any time in at least 500 years. Nonetheless, he asserted, that didn’t mean Hurricane Katrina, the sixth strongest Atlantic storm on record, had been brought on by climate change.

Even with a multitude of extreme weather events in recent years — tornadoes in New York City, blizzards in Washington, D.C., 15,000 warm-temperature records shattered across the U.S. in March — each consistent with computer models of a warming world, Emanuel and many other noted scientists have been unwilling to attribute any one event to global warming. There’s just too much variability in the weather, these experts say, and their dedication to data has helped prop open the door for “denialists” to sow doubt about the reality of our warming world.

But Hansen’s shot across the bow this morning indicates that the unwillingness to point fingers may be changing. According to a peer-reviewed paper Hansen has submitted to a leading scientific journal and made available to Time.com prior to publication, scientists can now state “with a high degree of confidence” that some extremely high temperatures are in fact caused by global warming, simply because they occur much more frequently than they used to. (A preliminary draft of the article is available here.)

Hansen’s reasoning has to do with math. Statisticians employ standard deviation to measure variability; it’s the calculation pollsters use to determine margin of error, and it’s especially valuable when looking at the weather. Perfect distribution of standard deviation is graphed as the familiar bell curve; about two-thirds of the time, data points fall in the middle of the bell — or within one standard deviation of the mean.

Hansen, with co-authors Reto Ruedy, also of NASA, and Makiko Sato, of Columbia University, has crunched decades’ worth of readings from more than 1,000 weather stations around the world as well as satellite observations and measurements from Antarctic research stations. The aim: to figure out how often temperatures varied from the mean — and how far they varied — during two periods.

In the paper, which Time.com confirmed has been peer-reviewed, the authors show that extreme outliers of more than three standard deviations above the mean temperature covered between six and thirteen percent of the globe during the years 2003 to 2008. If they were normally distributed and similar to the climactic record, that should have been just a 0.1-to-0.2 percent frequency of an extreme heat event. (That’s about exactly as often as a perfect bell curve predicts they would occur.) Hansen dubs this difference a “three-sigma anomaly,” for the Greek-letter symbol for standard deviation. And in the world of statistics, these anomalies represent a stunning 10-fold increase in extreme weather events.

Hansen says the heat wave that struck Texas and Oklahoma last summer and the Moscow heat wave of 2010 (which caused 11,000 deaths in the city) are examples of three-sigma anomalies. In a paper published last year in the Proceedings of the National Academy of Sciences, Stefan Rahmstorf and Dim Coumou of the Potsdam Institute for Climate Impact Research, in Germany, wrote that it was 80 percent probable that the Moscow heat wave had been caused by global warming.

“These three-sigma anomalies,” Hansen says, “we can now say are due to global warming.” But what about the extreme cold snaps climate-change deniers keep pointing to? Even with global warming, Hansen told Time.com in an email, there “is still a broad bell curve. In fact, it has become broader, which means there will still be times when a season is colder than average. When that happens [people] should not say, ‘What happened to global warming?’ It will still be there — they are just looking at natural variability.”

Back in 1988, when Hansen was among the first and most credible scientists to sound the alarm about global warming, he, Ruedy and several co-authors came up with the concept of “climate dice.” Imagine dice with two sides red (for hot), two sides blue (for cold) and two sides white (average temperatures). If you roll the dice, you’re equally likely to get any result. With continued emissions of greenhouse gas, however, the authors predicted that by the early 21st century, four of the sides would be red.

“The climate dice are loaded now, just as we said back in the 1980s that they would be,” Hansen wrote to Time.com. “People should be able to recognize the change, especially the increasingly extreme events. Don’t be surprised if there are more examples this summer.”